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These conclusions specifically apply to 2D simulations of Double Mach Reflections at one resolution. Also, note that the Clawpack implementation is fundamentally more efficient than that of the WENO implementation, which is designed as a more general large eddy simulation (LES) code. Overall, it seems that Clawpack method with the van Albada limiter yields a simultion that is on average about as computationally expensive as a simulation conducted with a properly setup 4th order bandwidth WENO-TCD method. This comparision of computational time is made by comparing a Clawpack 2nd order van Albada limited simulation at twice the resolution with that of a 4th order bandwidth optimized WENO-TCD method with the appropropriate amount of WENO for stability. Comparing the **whole** time of these mentioned configurations (for both the constant and variable viscosity and conductivity cases and slip and no slip BCs with and without the rotated frame of reference), shows that the 4th order WENO-TCD method can be anywhere from 2x faster to 2x slower than the Clawpack method with a mesh that is 2x finer in both the x and y directions. This wide variability is created by the TCD flagging criterion which is effected different for each configuration. The "optimal" choice of the flagging criterion for the fastest simulation which is still stable, is determined by factors such as the resolution, order of accuracy, stencil size, boundary conditions, viscosity and conductivity models, final timestep, and if the reflecting "wedge" is modeled using the ghost fluid method or in a rotated frame of reference. Also, if one looks at the graphics and uses only those simulations which do not show the "ringing" phenomenon, the Clawpack method is faster.

These conclusions are definitely also applicable to general 1D simulations, but may or may not be applicable to 3D simulations. The computational cost of halfing the cell size in the x,y, and z directions all together is significantly larger than with only two dimensions. With double Mach reflections (and possibily compressible mixing layers in general), for up to the finest resolutions encountered with these specific 2D simulations, Clawpack generally faster when one uses more WENO to ensure stability. At higher resolutions, WENO-TCD may be faster if the method does not continue to "ring" (see next paragraph), and the maximum amount of TCD can be used. In this ideal case, assumming it is stable, WENO would be used everywhere except at and very close to the shock waves. Also, these simulations at a Mach number of 4.5 are not representative of others, such as deflagration and subsonic mixing problems.

As shown by the performance tables and the graphical results, if WENO is used too sparingly ("pcurv" max large), then the WENO-TCD method "rings" or oscillates and shows signs of instability. When running the simulations where this occurred, it is seen that the numerical solution has additional restarts. Before the restart the current AMROC implementation senses an instability and artificially forces the CFL number to be greater than 1.5. Then the method restarts the time stepping with a smaller time step such that the CFL condition is satisfied. This, in the end of the computation, increases the total number of time steps, making the computation slower. For these simulations, one also finds "ripples" in the graphs.

By comparing the WENO-TCD results, one finds that the best combination of stability and high resolution, low numerical dissipation results is found when WENO is used all on and around the slipline. One possibility for this is that even though the "exact solution" of the Navier-Stokes equations is smooth, at the resolutions used in this simulations, the mesh could be too coarse for the wide stencil of the high order WENO-TCD methods. Therefore, across the slipline/mixing layer the gradients are large enough such that slipline/mixing layer behaves as a discontinuity. Also of interest is that all these difficulties discussed are more of an issue for the 7-point TCD rather than the 5-point TCD stencil. For the bandwidth optimized variant, the 7-point TCD stencil is used with the 4th order WENO method and the 5-point TCD stencil is used with the 2nd order WENO method.

-- JackZiegler^{?} - 26 Aug 2008

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